A Quasi-Newton method for physically-admissible simulation of Poiseuille flow under fracture propagation

Coupled hydro-mechanical processes are of great importance to numerous engineering systems, e.g., hydraulic fracturing, geothermal energy, and carbon sequestration. Fluid flow in fractures is modeled after a Poiseuille law that relates the conductivity to the aperture by a cubic relation. Newton's method is commonly employed to solve the resulting discrete, nonlinear algebraic systems. It is demonstrated, however, that Newton's method will likely converge to nonphysical numerical solutions, resulting in estimates with a negative fracture aperture. A Quasi-Newton approach is developed to ensure global convergence to the physical solution. A fixed-point stability analysis demonstrates that both physical and nonphysical solutions are stable for Newton's method, whereas only physical solutions are stable for the proposed Quasi-Newton method. Additionally, it is also demonstrated that the Quasi-Newton method offers a contraction mapping along the iteration path. Numerical examples of fluid-driven fracture propagation demonstrate that the proposed solution method results in robust and computationally efficient performance.